MathDB

A3/B5

Part of 2020 Princeton University Math Competition

Problems(3)

2020 PUMaC Algebra A3 / B5

Source:

1/1/2022
Let {x}=x−⌊x⌋\{x\} = x- \lfloor x \rfloor . Consider a function f from the set {1,2,...,2020}\{1, 2, . . . , 2020\} to the half-open interval [0,1)[0, 1). Suppose that for all x,y,x, y, there exists a zz so that {f(x)+f(y)}=f(z)\{f(x) + f(y)\} = f(z). We say that a pair of integers m,nm, n is valid if 1≤m,n≤20201 \le m, n \le 2020 and there exists a function ff satisfying the above so f(1)=mnf(1) = \frac{m}{n}. Determine the sum over all valid pairs m,nm, n of mn{m}{n}.
floor functionalgebra
2020 PUMaC Geometry A3 / B5

Source:

12/31/2021
Let ABCDABCD be a cyclic quadrilateral with circumcenter OO and radius 1010. Let sides ABAB, BCBC, CDCD, and DADA have midpoints M,N,PM, N, P, and QQ, respectively. If MP=NQMP = NQ and OM+OP=16OM + OP = 16, then what is the area of triangle △OAB\vartriangle OAB?
geometry
2020 PUMaC NT A3 / B5

Source:

1/1/2022
Find the sum (in base 1010) of the three greatest numbers less than 1000101000_{10} that are palindromes in both base 1010 and base 55.
number theory